Defining polynomial
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$( x^{2} + 2 x + 2 )^{6} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{5} + 3 ( x^{2} + 2 x + 2 )^{4} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{3})$: | $C_2$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[2]$ |
| Visible Swan slopes: | $[1]$ |
| Means: | $\langle\frac{2}{3}\rangle$ |
| Rams: | $(2)$ |
| Jump set: | undefined |
| Roots of unity: | $8 = (3^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.2.2.2a1.1, 3.2.3.8a5.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{2} + 2 x + 2 \)
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| Relative Eisenstein polynomial: |
\( x^{6} + 3 x^{4} + 3 t \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^3 + 2$,$2 z^2 + (2 t + 1)$ |
| Associated inertia: | $1$,$2$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $72$ |
| Galois group: | $C_2\times C_3^2:C_4$ (as 12T41) |
| Inertia group: | Intransitive group isomorphic to $C_3\times C_6$ |
| Wild inertia group: | $C_3^2$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[2, 2]$ |
| Galois Swan slopes: | $[1,1]$ |
| Galois mean slope: | $1.8333333333333333$ |
| Galois splitting model: | $x^{12} - 12 x^{9} + 26 x^{6} + 12 x^{3} + 1$ |